Finding the Assembly for Two Paths: Step-by-Step Guide

[asi123]

Homework Statement

Hey guys.
I have this two paths as you can see in the picture and I need to find their assembly (I hope I said it correctly).
Which one is correct, the right or the left?

Thanks.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Well, "assembly" isn't the correct English. "Union" of the two sets or "combination" of the two paths woud be better.

In any case, the problem, as I interpret this is to integrate some function from -R to R along the real line, then integrate from R to -R along the upper half circle with radius R. On the left, $\gamma1$ seems to be the line y= x or, in terms of complex numbers, t+ it, for t from -R to R. No, that is not at all what is given. But the picture on the right is not clear. You seem to be indicating that $\gamma1$ is raised up to some non-zero y, or in terms of complex numbers, t+ ai for some positive a. That is also not correct. $\gamma1$ is given as t+0i, not t+ some non-zero number times i. You should be showing $\gamma1$ running on the real axis, not above it.

 

[asi123]

Well, "assembly" isn't the correct English. "Union" of the two sets or "combination" of the two paths woud be better.

In any case, the problem, as I interpret this is to integrate some function from -R to R along the real line, then integrate from R to -R along the upper half circle with radius R. On the left, $\gamma1$ seems to be the line y= x or, in terms of complex numbers, t+ it, for t from -R to R. No, that is not at all what is given. But the picture on the right is not clear. You seem to be indicating that $\gamma1$ is raised up to some non-zero y, or in terms of complex numbers, t+ ai for some positive a. That is also not correct. $\gamma1$ is given as t+0i, not t+ some non-zero number times i. You should be showing $\gamma1$ running on the real axis, not above it.

Got you
So it actually the half circle over there together with the diameter, this is my path.

Thanks a lot and also thank you for the English correction

 

[asi123]

Well, the first part of the question ask me to find the integral in red (in the pic).
Is it right what I did?

 

[HallsofIvy]

What you have is correct but this looks more like a problem where you are expected to evaluate the integral around the closed path by using Residues. The integrand has poles of order 1 at i, -i, 2i, and -2i, of which i and 2i are inside the closed path.

 

[asi123]

What you have is correct but this looks more like a problem where you are expected to evaluate the integral around the closed path by using Residues. The integrand has poles of order 1 at i, -i, 2i, and -2i, of which i and 2i are inside the closed path.

Oh, yeah, you right, much easier.
And the points that are outside of the closed path equals to 0, right?

Thanks a lot.